## Electronic Communications in Probability

### Block size in Geometric($p$)-biased permutations

#### Abstract

Fix a probability distribution $\mathbf p = (p_1, p_2, \ldots )$ on the positive integers. The first block in a $\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\infty )$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with positive probability.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 80, 10 pp.

Dates
Accepted: 11 October 2018
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/euclid.ecp/1540346604

Digital Object Identifier
doi:10.1214/18-ECP182

Mathematical Reviews number (MathSciNet)
MR3873787

Zentralblatt MATH identifier
1398.05004

#### Citation

Cristali, Irina; Ranjan, Vinit; Steinberg, Jake; Beckman, Erin; Durrett, Rick; Junge, Matthew; Nolen, James. Block size in Geometric($p$)-biased permutations. Electron. Commun. Probab. 23 (2018), paper no. 80, 10 pp. doi:10.1214/18-ECP182. https://projecteuclid.org/euclid.ecp/1540346604

#### References

• [1] A. D. Barbour, L. Holst, and S. Janson, Poisson approximation, Oxford science publications, Clarendon Press, 1992.
• [2] Riddhipratim Basu and Nayantara Bhatnagar, Limit theorems for longest monotone subsequences in random Mallows permutations, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53 (2017), no. 4, 1934–1951.
• [3] J. Duchamps, J. Pitman, and W. Tang, Renewal sequences and record chains related to multiple zeta sums, ArXiv e-prints (2017), To appear in Transactions of the American Mathematical Society.
• [4] James Allen Fill, Hosam M. Mahmoud, and Wojciech Szpankowski, On the distribution for the duration of a randomized leader election algorithm, The Annals of Applied Probability 6 (1996), no. 4, 1260–1283.
• [5] Alexander Gnedin, Alexander Iksanov, and Alexander Marynych, Limit theorems for the number of occupied boxes in the Bernoulli sieve, 16 (2010).
• [6] Alexander Gnedin, Alexander Iksanov, and Alexander Marynych, A generalization of the Erdős–Turán law for the order of random permutation, Combinatorics, Probability and Computing 21 (2012), no. 5, 715–733.
• [7] Alexander Gnedin and Grigori Olshanski, $q$-Exchangeability via quasi-invariance, The Annals of Probability (2010), 2103–2135.
• [8] Alexander V Gnedin et al., The Bernoulli sieve, Bernoulli 10 (2004), no. 1, 79–96.
• [9] Alexander V Gnedin, Alexander M Iksanov, Pavlo Negadajlov, Uwe Rösler, et al., The Bernoulli sieve revisited, The Annals of Applied Probability 19 (2009), no. 4, 1634–1655.
• [10] Svante Janson and Wojciech Szpankowski, Analysis of an asymmetric leader election algorithm, the electronic journal of combinatorics 4 (1997), no. 1, 17.
• [11] Alexander Marynych, Alexander Iksanov, and Alexander Gnedin, The Bernoulli sieve: an overview, Discrete Mathematics & Theoretical Computer Science (2010).
• [12] Jim Pitman and Wenpin Tang, Regenerative random permutations of integers, ArXiv e-prints: 1704.01166 (2017), To appear in Annals of Probability.